Theorem pjo

Description: The orthogonal projection. Lemma 4.4(i) of Beran p. 111. (Contributed by NM, 30-Oct-1999) (New usage is discouraged.)

	Ref	Expression
Assertion	pjo	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pjch1	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 )
2	1	adantl	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) = 𝐴 )
3		axpjpj	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → 𝐴 = ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) )
4	2 3	eqtr2d	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) )
5		helch	⊢ ℋ ∈ Cℋ
6	5	pjcli	⊢ ( 𝐴 ∈ ℋ → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ )
7	6	adantl	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ )
8		pjhcl	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ )
9		choccl	⊢ ( 𝐻 ∈ Cℋ → ( ⊥ ‘ 𝐻 ) ∈ Cℋ )
10		pjhcl	⊢ ( ( ( ⊥ ‘ 𝐻 ) ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ )
11	9 10	sylan	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ )
12		hvsubadd	⊢ ( ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ∈ ℋ ∧ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ∈ ℋ ∧ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ∈ ℋ ) → ( ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ↔ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) )
13	7 8 11 12	syl3anc	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ↔ ( ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) +ℎ ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) ) )
14	4 13	mpbird	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) = ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) )
15	14	eqcomd	⊢ ( ( 𝐻 ∈ Cℋ ∧ 𝐴 ∈ ℋ ) → ( ( projℎ ‘ ( ⊥ ‘ 𝐻 ) ) ‘ 𝐴 ) = ( ( ( projℎ ‘ ℋ ) ‘ 𝐴 ) −ℎ ( ( projℎ ‘ 𝐻 ) ‘ 𝐴 ) ) )